In the expression below if a = 4, what value of b is needed for the expression to have a value of 81?
12a^3 b^7∙(ab)^(-4)
A cube has side lengths of 25 cm long. If Sam has 16 of these cubes, what total volume would the cubes occupy written as a power of 2?
The total volume is 2____ cm3
To find the value of b in the expression 12a^3 b^7∙(ab)^(-4) when a = 4 and the expression has a value of 81, we can substitute the values and solve for b.
Substituting a = 4: 12(4)^3 b^7∙(4b)^(-4) = 81
Simplifying: 12(64) b^7∙(4b)^(-4) = 81
Simplifying further: 768 b^7∙(4b)^(-4) = 81
Dividing both sides by 768: b^7∙(4b)^(-4) = 81/768
Simplifying the right side: b^7∙(4b)^(-4) = 1/9
To simplify the expression, we can write (4b)^(-4) as 1/(4b)^4:
b^7∙1/(4b)^4 = 1/9
Multiplying both sides by (4b)^4: b^7 = (4b)^4/9
Expanding the denominator on the right side: b^7 = 64b^4/9
Multiplying both sides by 9 to eliminate the fraction: 9b^7 = 64b^4
Dividing both sides by b^4: 9b^7/b^4 = 64
Simplifying the left side: 9b^(7-4) = 64
Simplifying further: 9b^3 = 64
Dividing both sides by 9: b^3 = 64/9
Taking the cube root of both sides: b = (64/9)^(1/3)
Therefore, the value of b needed for the expression to have a value of 81 when a = 4 is approximately 2.075.